Here is a calendar of
Chicago sporting events.
I know a lot of bars/restaurants show various international sports, but I'm not well connected to that world so I can't point you to any specifics. You might want to email the Italian consulate (which, if I remember correctly, is located just on Michigan Ave. or very close to it)--I'm sure they could point you to a few locations.
The
Green Mill is one of Chicago's most legendary jazz clubs.